Short intervals for the Romanoff-type sumset

TL;DR

This paper demonstrates that almost all short intervals of a certain length within a large range contain approximately the expected number of integers that are sums of a prime and an element from a lacunary set generated by powers of two, with high probability.

Contribution

It establishes a new result on the distribution of sums of primes and lacunary sets in short intervals, extending understanding of additive number theory for these special sumsets.

Findings

Almost all short intervals contain the expected number of prime-plus-lacunary set integers.

The result holds for intervals of length between X^{2/15+ε} and X^{0.99}.

The number of exceptions is exponentially small in a power of log X.

Abstract

Let $X$ be large and let $\mathcal{P}$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda\geqslant 1$ determined by a balancing relation, and let $\mathcal{A}_{\lambda}(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\mathcal{S}_{\lambda}:=\mathcal{P}+\mathcal{A}_{\lambda}(X)$. Fix $\varepsilon>0$, choose $\theta$ with $2/15+\varepsilon<\theta<0.99$, and set $h=X^{\theta}$. We prove that for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$, where $p$ is prime and $a\in\mathcal{A}_{\lambda}(X)$.